REGULARITY AND CONVERGENCE OF LOCAL FIRST INTEGRALS OF ANALYTIC DIFFERENTIAL SYSTEMS

The paper states and sketches proofs of Theorem 1.1 (C∞ first integral exists iff the origin is a nonisolated singularity) and Theorem 1.2 (a pluripolar/all dichotomy in finite-dimensional parameter families) . In the sufficiency direction of Theorem 1.1, the proof crucially invokes a stable/unstable manifold decomposition by asserting that (1.2) implies all eigenvalues of the transverse block B have nonvanishing real parts, then uses a homological equation solved by an integral along the flow (equations (2.6)–(2.11)) to eliminate a flat remainder . However, the resonance condition (1.2) does not imply Re(λj) ≠ 0, so the needed hyperbolicity is not guaranteed; without it, convergence of the integral construction is not justified. The model’s solution mirrors the same normalization and C∞ conjugacy strategy (distinguished Poincaré–Dulac, Borel’s lemma, then an integral solution of the homological equation along the transverse linear flow) and likewise omits the hyperbolicity requirement, leaving the sufficiency part incomplete as well. By contrast, the necessity argument and the parameter-space dichotomy (via Lemma 1 on polynomial parameter dependence of the formal coefficients and the Bernstein–Walsh lemma in Lemma 2) align with the paper and are sound within their scope .